On the Connectedness of Random Sets of ℝ

Abstract

Random sets arise as distinguished models to study probability under imprecision. This work aims to determine the connectedness of a random set in terms of its capacity functional (the probability of hitting a set). This problem is linked to the celebrated Choquet–Kendall–Matheron theorem, which states that a random closed set is characterized by its capacity functional. Hence, such functional must also characterize any topological property of a random set, such as its connectedness. While the capacity functional of a random closed set operates over the large class of compact sets, we show that the determination of connectedness can be checked only over a simple family of sets.